1. Adding and Subtracting Real Numbers Section 1-5

2. Goals Goal To find sums and differences of real numbers. Rubric Level 1 – Know the goals. Level 2 – Fully understand the goals. Level 3 – Use the goals to solve simple problems. Level 4 – Use the goals to solve more advanced problems. Level 5 – Adapts and applies the goals to different and more complex problems.

3. Vocabulary Absolute value Opposite Additive inverses

4. The set of all numbers that can be represented on a number line are called real numbers. You can use a number line to model addition and subtraction of real numbers. Addition To model addition of a positive number, move right. To model addition of a negative number, move left. Subtraction To model subtraction of a positive number, move left. To model subtraction of a negative number, move right. Real Numbers

5. Add or subtract using a number line. Start at 0. Move left to –4. 1110 98 7 654321 0 + (–7) –4 + (–7) = –11 To add –7, move left 7 units. –4 –4 + (–7) Example: Adding & Subtracting on a Number Line

6. Add or subtract using a number line. Start at 0. Move right to 3. To subtract –6, move right 6 units. -3-20123456 7 89 + 3 3 – (–6) = 9 3 – (–6) – ( – 6) Example: Adding & Subtracting on a Number Line

7. Add or subtract using a number line. –3 + 7 Start at 0. Move left to –3. To add 7, move right 7 units. -3 -2 01234 5 6 7 89 –3 +7 –3 + 7 = 4 Your Turn:

8. Add or subtract using a number line. –3 – 7 Start at 0. Move left to –3. To subtract 7, move left 7 units. –3–3 –7–7 11 10 987 6 54 3210 –3 – 7 = –10 Your Turn:

9. Add or subtract using a number line. –5 – (–6.5) Start at 0. Move left to –5. To subtract –6.5, move right 6.5 units. 87 65 43 210 –5 –5 – (–6.5) = 1.5 1 2 – ( – 6.5) Your Turn:

10. Definition Absolute Value – The distance between a number and zero on the number line. – Absolute value is always nonnegative since distance is always nonnegative. –The symbol used for absolute value is | |. Example: –The |-2| is 2 and the |2| is 2.

11. The absolute value of a number is the distance from zero on a number line. The absolute value of 5 is written as |5|. 5 units 2101 234 56 6543 -- - -- - |5| = 5|–5| = 5 Absolute Value on the Number Line

12. Rules For Adding

13. Add. Use the sign of the number with the greater absolute value. Different signs: subtract the absolute values. A. B.–6 + (–2) (6 + 2 = 8) –8–8 Both numbers are negative, so the sum is negative. Same signs: add the absolute values. Example: Adding Real Numbers

14. Add. –5 + (–7) –12 Both numbers are negative, so the sum is negative. Same signs: add the absolute values. a. (5 + 7 = 12) – 13.5 + ( – 22.3) b. (13.5 + 22.3 = 35.8) – 35.8 Both numbers are negative, so the sum is negative. Same signs: add the absolute values. Your Turn:

15. c. 52 + (–68) (68 – 52 = 16) –16 Use the sign of the number with the greater absolute value. Different signs: subtract the absolute values. Add. Your Turn:

16. Definition Additive Inverse – The negative of a designated quantity. – The additive inverse is created by multiplying the quantity by -1. Example: –The additive inverse of 4 is -1 ∙ 4 = -4.

17. Opposites Two numbers are opposites if their sum is 0. A number and its opposite are additive inverses and are the same distance from zero. They have the same absolute value.

18. Additive Inverse Property

19. Subtracting Real Numbers To subtract signed numbers, you can use additive inverses. Subtracting a number is the same as adding the opposite of the number. Example: –The expressions 3 – 5 and 3 + (-5) are equivalent.

20. A number and its opposite are additive inverses. To subtract signed numbers, you can use additive inverses. 11 – 6 = 5 11 + (–6) = 5 Additive inverses Subtracting 6 is the same as adding the inverse of 6. Subtracting a number is the same as adding the opposite of the number. Subtracting Real Numbers

21. Rules For Subtracting

22. Subtract. –6.7 – 4.1 –6.7 – 4.1 = –6.7 + (–4.1) To subtract 4.1, add –4.1. Same signs: add absolute values. –10.8 Both numbers are negative, so the sum is negative. (6.7 + 4.1 = 10.8) Example: Subtracting Real Numbers

23. Subtract. 5 – ( – 4) 5 − (–4) = 5 + 4 9 To subtract –4, add 4. Same signs: add absolute values. (5 + 4 = 9) Both numbers are positive, so the sum is positive. Example: Subtracting Real Numbers

24. On many scientific and graphing calculators, there is one button to express the opposite of a number and a different button to express subtraction. Helpful Hint

25. Subtract. 13 – 21 To subtract 21, add – 21. Different signs: subtract absolute values. Use the sign of the number with the greater absolute value. –8 = 13 + (–21) (21 – 13 = 8) Your Turn:

26. –14 – (–12) Subtract. –14 – (–12) = –14 + 12 (14 – 12 = 2) To subtract –12, add 12. Use the sign of the number with the greater absolute value. –2 Different signs: subtract absolute values. Your Turn:

27. An iceberg extends 75 feet above the sea. The bottom of the iceberg is at an elevation of –247 feet. What is the height of the iceberg? Find the difference in the elevations of the top of the iceberg and the bottom of the iceberg. elevation at top of iceberg minus elevation at bottom of iceberg 75 – (–247) 75 – (–247) = 75 + 247 = 322 To subtract –247, add 247. Same signs: add the absolute values. – 75 – 247 Example: Application The height of the iceberg is 322 feet.

28. What if…? The tallest known iceberg in the North Atlantic rose 550 feet above the ocean's surface. How many feet would it be from the top of the tallest iceberg to the wreckage of the Titanic, which is at an elevation of –12,468 feet? elevation at top of iceberg minus elevation of the Titanic – 550 – (–12,468) 550 – (–12,468) = 550 + 12,468 To subtract –12,468, add 12,468. Same signs: add the absolute values. = 13,018 550 – 12,468 Your Turn: Distance from the top of the iceberg to the Titanic is 13,018 feet.

29. Joke Time What’s brown and sticky? A stick. What happened when the wheel was invented? It caused a revolution. Why was the calendar depressed? Because it’s days were numbered.

30. Assignment 1.5 Exercises Pg. 41 – 43: #10 – 76 even